For a military rifle, it was six of one, half dozen of another. The oldest reason I can remember reading was that cock on closing was there to reduce the extraction effort when the rifle got hot, and the ammunition was hot, leading to sticking extraction. I don't know if that is the real reason, or a made up reason. The M1917 Enfield mechanism half cocks the striker on opening. Closing provides the final effort to pull the firing pin back on cam down.
All my small ring mausers are cock on closing
1891 Argentine, cock on closing
Even on a cock on closing mechanism, the firing pin is retracted enough to free the firing pin tip from the primer.
You will find firing pin retraction cams are essential in all rotary bolt designs. Manufacturing tolerances then and now always allowed a certain amount of firing pin off set. Given a bolt that rotates in a circle, the firing pin tip can be broken if the primer indent hole is a little off center. So, the firing pin is always cammed enough to pull the firing pin behind the bolt face.
you see a firing pin retraction cam by turning a M1 carbine receiver up side down
The firing pin is pulled back on a K31 Schmidt Rubin before the lugs rotate
My theory on why cock on closing is not found on commercial rifles has to do with over ride triggers.
The pre 64 M70 has one of the best, and is the earliest over ride trigger I know of, but surely, there are others even earlier.
This is a Timney trigger
These over ride triggers are like trap door mechanisms. You kick away the prop, and the door falls. Notice on the Timney mechanism just how little sear surface engagement there is holding everything together. While I got used to two stage military triggers, and prefer two stage triggers on my target rifles, most shooters hate having an initial take up. So the single stage over ride trigger was developed. You can get a fine and light trigger pull if everything is set up just right. One of the things that will cause an over ridge trigger to "follow" is the bolt cocking piece hitting the trigger sear too hard. A hard hit on the sear will over ride the mechanism and cause the sear to drop.
Used to see that all the time on bolt gunners who adjusted their triggers very light. Oh it worked great in slow fire when the bolt was eased down, but in rapid fire, where the bolt is a flying, a minimum trigger pull weight would result in an alibi. The shooter would be racking that bolt fast, the cocking piece would hit that sear hard, the sear then tripped, the firing pin would go forward, and then the shooter wondering why his gun no go bang.When it happened to me, I always found a slight indentation on the primer, and always wondered why the primer did not go off. And was scared by that indentation.
A cock on closing mechanism would beat the heck out of the typical over ride trigger mechanism. And, you will probably bruise your hand on your scope bell closing the thing. You have to push, remember?
I had the bolt handle lengthened on this M1917, because the handle that Joe Donahue had installed, (he was the first owner) caused my hand to hit the scope bell. And that hurt.